Quadrilateral Inscribed in a Quadrilateral
In both of my constructions I began by constructing the interior quadrilateral to the requirements suggested in the question. I then constructed the exterior quadrilateral so that it could be manipulated. After manipulating both figures, I found that the interior quadrilateral stayed true and did so no matter what changes I made to the exterior figure.
I also found that regardless of the manipulations in either construction it would never result in a non-convex interior figure. The exterior figure could be changed to be non-convex, but never the interior.
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